FORMATION OF DISKS AND BINARIES
P. BODENHEIMER
UCO/Lick Observatory
University of California, Santa Cruz, CA 95064, USA
HAROLD W. YORKE
Jet Propulsion Laboratory
Pasadena, CA 91109, USA
AND
ANDREAS BURKERT
Max-Planck-Institute for Astronomy
D-69117 Heidelberg, Germany
1.
Introduction
Two possible outcomes are considered for the collapse of a rotating core
of a molecular cloud. For the rst case, two-dimensional axisymmetric hydrodynamical collapse calculations with radiative transfer are presented to
study the formation of a central star and a disk. For the second case, threedimensional calculations provide insight on the process of the formation of
binary and multiple systems of protostars. In general when a protostar or
fragment collapses it passes through three phases. The rst, the isothermal
phase, corresponds to low-density gas which is optically thin to infrared
radiation. The second, adiabatic, phase starts in the central regions of the
protostar or fragment when it becomes dense enough to be optically thick;
thus as it compresses it heats. The third, accretion, phase starts when the
central regions of the collapsing fragment reach hydrostatic equilibrium and
form the core of a star. Remaining infalling material accretes onto this central object, either directly or through a disk. In the case of the axisymmetric
solutions presented here, the evolution passes through all three phases. The
3-D simulations, however, are limited mainly to the isothermal phase.
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Simulations of Disk Formation
A number of important issues should be considered in this connection,
many of which are still unresolved. (1) What are the initial conditions for
the formation of a star-disk system, as opposed to a binary? (2) What is the
predicted appearance of the emergent spectrum during the early phases of
disk formation and evolution? (3) What are the important mechanisms for
angular momentum transport in a disk? At what stage is each dominant?
What are the time scales? (4) What are the implications of the angular
momentum transport process with regard to planet formation? (5) Will a
gravitationally unstable disk formed from cloud collapse fragment, or will
it transfer mass to the central object through the action of gravitational
torques?
2.1.
ASSUMPTIONS AND METHOD
Detailed simulations of the collapse of rotating clouds with radiation transport have previously been published by Yorke, Bodenheimer, and Laughlin
(1993, 1995). The example presented here is based on a nested{grid computer code with improved physics and longer evolutionary time scale as
compared with the earlier work. The code includes two-dimensional Eulerian hydrodynamics, radiation transport in the ux-limited diusion approximation, no magnetic elds, and a radiative opacity based on the properties of dust grains, including silicates, ice-coated silicates, and carbon.
When the disk becomes gravitationally unstable, as indicated by a minimum Toomre Q value of 1.5 or less, ordinary viscosity, based on the {law
prescription, is included according to the procedure described in Roz_ yczka,
Bodenheimer, & Bell (1994) and Laughlin & Bodenheimer (1994). However in this case we assume that the gravitational instability results in
disk evolution and angular momentum transport such that the condition of
marginal gravitational instability is maintained. Thus we adjust the value
of so that the minimum Q value remains between 1.3 and 1.5.
The innermost grid point has a size of a few AU; thus the central stellar
core is not resolved. This region is modelled as a central star plus disk
with luminosity equal to the sum of the contraction luminosity of the star
and the accretion luminosity of disk onto star. This luminosity serves as an
inner boundary condition for the radiation transfer in the grid. At selected
times, frequency-dependent radiation transfer is carried out so that the
emergent infrared spectrum and isophotal contours of the protostar can be
determined. A ray-tracing solution of the transfer equation is obtained on
a grid of lines of sight through the object, at 64 frequencies, including true
absorption and scattering.
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FORMATION OF DISKS AND BINARIES
2000
10
10
3 km s-1
DISTANCE FROM EQUATOR
20
1000
20
10
0
1000
2000
3000
2000
1000
1000
0
DISTANCE FROM AXIS [ AU ]
2000
3000
4000
Disk structure in a protostar of 2 M after a time of 585000 yr. (Right):
Contours of equal density and velocity vectors in a meridional plane. (Left): Contours of
equal gas temperature. The pair of thick lines indicate one density scale height above the
equatorial plane. From Yorke & Bodenheimer (1998).
Figure 1.
2.2.
RESULTS
One example will be presented here; further details may be found in Yorke
& Bodenheimer (1998) and Yorke & Kaisig (1995). The initial condition is
a uniformly rotating cloud of 2 M , radius 2 21017 cm, density distribution
/ r02 . The initial ratios of thermal and rotational energies, respectively,
to the absolute value of the gravitational potential energy are = 0:39
and = 0:008. The specic angular momentum R2
at the outer edge is
4 21021 cm2 s01; all of these values are consistent with observations. The
calculation is carried out on four xed Eulerian nested grids, 124 by 124
zones each, with a factor 2 increase in resolution per grid. On the innermost
grid the zone size is 2 21014 cm.
During most of the evolution the protostar is divided into three regions:
the unresolved stellar core, the equilibrium disk, and the infalling envelope.
An example of the disk structure is shown in Figure 1. Outer and inner
accretion shocks are evident interior to a radius of 1000 AU. The disk
becomes gravitationally unstable, and mass accretes onto the central star
at a typical rate of 2 21006 M yr01 , generating a luminosity of 10
L . The outer disk radius expands considerably with time as a result of
angular momentum transport and accretion. After a protostellar lifetime (a
few 2105 yr) the disk mass has decreased to about 35% of the total mass
and the accretion rate has also dropped considerably. Thus the calculated
disk mass is higher than those observed in young stellar objects, and we
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P. BODENHEIMER ET AL.
conclude that some process of angular momentum transport, other than
gravitational instability, must also operate during the early phases of disk
evolution.
The infrared spectrum of the model shown in Figure 1 is strongly dependent on viewing angle. By this time most of the infalling material along
the polar axis has collapsed onto the central star, the dust optical depth is
low, and if one observes from this direction one can detect the central object
and inner disk. Some optical radiation is present, and the spectrum is fairly
at in a [log , log(L )] diagram from 0.8 to 20 m. On the other hand,
if one views in the equatorial plane, the central object is highly obscured
and the total ux is about a factor 30 less than that observed pole-on. The
spectrum has two peaks, one at about 100 m, and another, smaller one in
the optical and near infrared. This second peak is caused by scattered light
from the central star and inner disk. We conclude that infrared searches for
protostars are biased in the sense that pole-on objects are the most likely
ones to be discovered.
3.
Binary Formation by Fragmentation
The general goal of three-dimensional calculations is to explain the observed properties of binary systems, such as their frequency and their distributions of periods, secondary masses, and eccentricities. A number of
formation mechanisms have been discussed (reviewed by Boss 1993a and
Bodenheimer et al 1993), including fragmentation during protostar collapse, disk fragmentation, cloud{cloud collisions, capture, and ssion. Here
we consider only the rst of these processes.
Various physical assumptions have been employed in collapse calculations, including an isothermal equation of state, an equation of state with
cooling, an adiabatic or polytropic equation of state, and radiative transfer
in the ux-limited diusion approximation or the Eddington approximation. Typical initial conditions include the following assumptions:
Cloud has an assumed geometry (spherical, or cylindrical, or oblate, or
prolate
Cloud has no infall motion and is rotating uniformly
Cloud has a mass of about 1 solar mass and a radius of 0.05 parsec
Cloud has a distribution of density that is somewhat centrally condensed,
in agreement with observations of cores of molecular clouds
Cloud has an initial assumed density perturbation, for example random
noise or a 10% m = 2 mode
FORMATION OF DISKS AND BINARIES
127
The typical 3-D hydro calculations show the formation of a binary system on an eccentric orbit with separation in the range 10{100 AU, or a
multiple system on the same scale. For example, Boss (1996) and Klapp
& Sigalotti (1998) have found systems with 5{10 small fragments. Miyama
(1992) did adiabatic collapse calculations and found agreement with the analytical estimate for clouds of initial uniform density (with = 7=5), which
shows that fragmentation should take place only if < 0:09 0:2 . Bonnell
& Bate (1994), in an isothermal SPH calculation, saw the formation of an
initial binary of unequal masses, evolution toward equal masses, and then
formation of a second pair of fragments as a result of the spiral waves induced by the original binary. The results of many of the early fragmentation
calculations need to be reexamined, because the numerical resolution was
too coarse to satisfy the Jeans condition discussed by Truelove et al (1997).
Rather than attempt to review a wide variety of other results, we focus on
four specic questions:
1. A typical observed molecular cloud core is centrally condensed and
has 0:4, 0:01. Will it fragment?
2. Does fragmentation occur from an initial power{law density distribution (say / r01) with uniform rotation?
3. Do nite-dierence codes and SPH codes give the same results on the
same fragmentation problem?
4. Do fragmentation results depend strongly on the form and amplitude
of the assumed initial perturbation?
The answer to the rst question is most clearly illustrated by a series of
results by Boss (1993b). His collapse calculations, which include radiation
transport, start from a Gaussian prolate density distribution with a ratio of
central density to edge density of about 20. A random density perturbation
is imposed. A set of models is constructed, corresponding to various points
in the (; ) plane to determine which clouds fragment and which remain as
single objects. If the ratio of axes in the prolate cloud is 2:1, the cloud with
the given initial condition fragments. If the ratio, on the other hand, is 1.5:1,
the cloud collapses but remains single. Thus the fate of a \standard" cloud
may depend sensitively on initial conditions and the numerical technique,
and further studies of this case should be done.
The second and third questions were considered by Burkert, Bate, &
Bodenheimer (1997). In connection with the second question it is important
to know what the actual degree of central concentration of a cloud is, just
before it begins to collapse. Observationally, recent work by Andre, WardThompson, & Motte (1996) shows that pre-collapse cloud cores have density
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P. BODENHEIMER ET AL.
distributions close to / r02 in the outer parts, but a signicant fraction
of the inner part of the cloud has / r01 . Theoretically, calculations of
the evolution of clouds, as controlled mainly by ambipolar diusion, before
collapse (Tomisaka, Ikeuchi, & Nakamura 1990; Lizano & Shu 1989; Ciolek
& Mouschovias 1994; Basu & Mouschovias 1994) show that the density
distribution just before collapse starts is quite close to / r02. With such
a degree of central condensation and with uniform rotation (which is likely
because of magnetic braking) fragmentation does not appear to occur. But
what about the r01 law? Burkert et al (1997) were able to show that, with
this initial density distribution, with an initial density perturbation of the
form m = 2 and a 10% amplitude, and with = :25; = :23; fragmentation
into four objects occurs in the inner region of the cloud, on a scale of 100
AU. The calculation was done with an Eulerian grid code, with xed nested
grids such that the resolution on the innermost grid was 1003R, where R
is the total radius of the cloud. The same problem was calculated by use
of an SPH code with 200 000 particles. The resolution element in such a
calculation is the smoothing length, which varies through the volume; in
this case it was adjusted to be 1003R at a density of 10012 g cm03, which is
about the point where fragmentation starts. However the spatial resolution
of the two codes can be quite dierent at other densities. Nevertheless, the
fragmentation observed in the SPH calculation was very similar to that in
the grid code. Only dierences in small details were observed (see Burkert et
al 1997). One can conclude that a typical observed density distribution in a
cloud core is unstable to fragmentation, from which one can understand the
fact that a large fraction of young stars are observed to be in binary systems.
Note, however, that the actual simulations were carried out for values of and that are somewhat dierent from those typically observed.
For the answer to the fourth question we report on the results of some
new calculations. The initial cloud had a mass of 1 M , a radius of 5 21016
cm, isothermal temperature of 10 K, and a centrally condensed Gaussian
density distribution, corresponding to = 0:26; = 0:16. The main grid
has 64 2 64 2 64 Cartesian zones. Embedded in it are three subgrids with
128 2 128 2 64 zones in the x; y; and z directions, respectively. The x{values
of the outer edges of the subgrids were 2.5 21016 ; 6:25 2 1015, and 1:56 2 1015
cm. The resolution on the innermost grid is R=2000. The same conguration was run with three dierent initial density perturbations. The rst run
had no initial perturbation, other than that induced by the grid. A Fourier
analysis of the growth of the modes in the azimuthal direction shows that
the m = 4 mode grows most rapidly, followed closely by m = 8. The m = 1
and m = 2 perturbations stay at a very low level. Thus fragmentation is
likely to be numerically induced. The result is shown in the left panel of
Figure 2. A high-density condensation forms in the very center, then an
FORMATION OF DISKS AND BINARIES
129
Gray-scale representation of the density distribution in the equatorial plane
of a fragmenting cloud (left) with no initial density perturbation and (right) with an
initial
= 2 perturbation of 10% amplitude. Black represents highest density. The box
dimensions are 1.5 21015 by 1.5 21015 cm.
Figure 2.
m
axisymmetric ring builds up around it, and the ring fragments into eight
symmetrically located pieces. At the time corresponding to the gure the
symmetry of the ring has just begun to be broken as the fragments in the
unstable conguration gravitationally interact. The result is quite dierent
if an initial m = 2 perturbation with 10% amplitude is applied. The Fourier
analysis shows that the m = 2 amplitude remains almost constant in time,
at 10%. The m = 4 mode, which is probably at least in part numerical,
grows quite rapidly but remains at least an order of magnitude in amplitude
below that for m = 2. Other modes remain at a low level until the fragmentation begins, at which time all modes increase rapidly in amplitude.
The result is shown in the right panel of Figure 2. A central condensation
forms, and the initial bar-like perturbation wraps up into a spiral pattern,
generating two additional fragments in the process. Further details for this
case can be found in Burkert & Bodenheimer (1996). The third case was
run with a random initial density perturbation such that the typical mode
started with an amplitude of 1003 { 1004. Initially the modes grow slowly,
with m = 1 and m = 2 dominating; later all modes grow rapidly with
m = 3 slightly ahead of the rest. The result, which is not shown, is similar
to that for the rst run, except that the ring fragments into three pieces
rather than 8. Thus the results at the time of initial fragmentation are quite
dierent for the three runs with dierent initial perturbations. However no
conclusions can be drawn concerning the longer-term evolution.
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P. BODENHEIMER ET AL.
Conclusion
Some general conclusions can be summarized concerning the 3-D collapse
results.
1. Fragmentation calculations have many possible outcomes, including
binary formation, fragmentation in a surrounding disk induced by an initial
binary, formation of a small cluster, formation of laments which fragment,
or the formation of a binary plus low-mass fragments.
2. No fragmentation calculation has been carried out long enough so that
most of the material in the original cloud has collapsed to the equatorial
plane. Not all of this material will eventually end up in fragments, but
the nal outcome in these systems, as inuenced by captures, mergers,
accretion, and escape of gas and stars, is still not known.
3. The results may depend qualitatively on numerical resolution, particularly if the Jeans condition is not satised. Calculations must show
numerical convergence. The initial perturbation has a strong eect on the
initial phase of fragmentation.
4. An initially highly centrally condensed cloud with uniform rotation
fragments in the case / r01 , but the situation for steeper power laws is
still unclear.
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